Question

A test consists of ‘True’ or ‘False’ questions. One mark is awarded for every correct answer while \[\dfrac{1}{4}\] mark is deducted for every wrong answer. A student knew answers to some of the questions. Rest of the questions he attempted by cheating. He answered 120 questions and got 90 marks. If the answers to all questions he attempted by cheating were wrong, then how many questions did he answer correctly? How the habit of cheating will affect his character building?

Answer 1

Hint: Here first we will assume the number of questions attempted by cheating as x and then we will find the number of questions attempted correctly and then we will form a linear equation and then solve for the value of x to get the desired answer.

Complete step-by-step answer:
Let the number of questions attempted by cheating be x.
Now we are given the total number of questions attempted are 120
Therefore, the number of questions attempted correctly are given by:-
 \[{\text{number of questions attempted correctly}} = {\text{total questions}} - {\text{number of questions attempted by cheating}}\]
Putting in the known values we get:-
\[{\text{number of questions attempted correctly}} = 120 - x\]
Now it is given that one mark is awarded for every correct answer while \[\dfrac{1}{4}\] mark is deducted for every wrong answer
Also, the student scored 90 marks.
Hence we will form linear equation such that:
\[{\text{marks scored = }}\left( {{\text{marks awarded for every correct answer}}} \right) \times \left( {{\text{number of questions attempted correctly}}} \right) - \left( {{\text{marks deducted by each wrong answer}}} \right) \times \left( {{\text{number of questions attempted by cheating}}} \right)\]
Putting in the respective values we get:-
\[90 = \left( 1 \right) \times \left( {120 - x} \right) - \left( {\dfrac{1}{4}} \right) \times \left( x \right)\]
Solving it further we get:-
\[90 = \left( {120 - x} \right) - \left( {\dfrac{x}{4}} \right)\]
Now taking the LCM we get:-
\[90 = \dfrac{{4\left( {120 - x} \right) - x}}{4}\]
Solving it further we get:-
\[90 = \dfrac{{480 - 4x - x}}{4}\]
\[ \Rightarrow 90 = \dfrac{{480 - 5x}}{4}\]
Now cross multiplying it we get:-
\[90 \times 4 = 480 - 5x\]
\[ \Rightarrow 360 = 480 - 5x\]
Now solving for the value of x we get:-
\[5x = 480 - 360\]
\[ \Rightarrow 5x = 120\]
Dividing the equation by 5 we get:-
\[x = \dfrac{{120}}{5}\]
\[ \Rightarrow x = 24\]

Hence the number of questions attempted by cheating is 24.

Note: Students might make mistakes in forming the linear equation. They should note that we have to multiply the marks awarded or deducted with the respective number of questions attempted correctly or by cheating.
A linear equation in one variable is an equation in which the highest power of the variable is one and has only one variable and is of the form:
\[ax + b = 0\]